The derivative of `f(x) = e^(2x)` is

Options

A. `2 e^(2x)`
B. `e^(2x)`
C. `(e^(2x))/(2)`
D. `- e^(2x)`

Answer

To find the derivative of the function \( f(x) = e^{2x} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the function**: We have the function \( f(x) = e^{2x} \).

2. **Apply the chain rule**: The derivative of an exponential function \( e^{u} \) with respect to \( x \) is given by \( e^{u} \cdot \frac{du}{dx} \), where \( u = 2x \) in this case.

3. **Differentiate the exponent**: We need to find \( \frac{du}{dx} \) where \( u = 2x \). The derivative of \( 2x \) with respect to \( x \) is \( 2 \).

4. **Combine the results**: Now, applying the chain rule:
\[ f'(x) = e^{2x} \cdot \frac{du}{dx} = e^{2x} \cdot 2 \]

5. **Final result**: Therefore, the derivative of \( f(x) = e^{2x} \) is:
\[ f'(x) = 2e^{2x} \]

### Summary:
The derivative of \( f(x) = e^{2x} \) is \( f'(x) = 2e^{2x} \).